Pattern formation and phase transition in the collective dynamics of a binary mixture of polar self-propelled particles.

The collective behavior of a binary mixture of polar self-propelled particles (SPPs) with different motile properties is studied. The binary mixture consists of slow-moving SPPs (sSPPs) of fixed velocity v_{s} and fast-moving SPPs (fSPPs) of fixed velocity v_{f}. These SPPs interact via a short-range interaction irrespective of their types. They move following certain position and velocity update rules similar to the Vicsek model (VM) under the influence of an external noise η. The system is studied at different values of v_{f} keeping v_{s}=0.01 constant for a fixed density ρ=0.5. Different phase-separated collective patterns that appear in the system over a wide range of noise η are characterized. The fSPPs and the sSPPs are found to be orientationally phase synchronized at the steady state. We studied an orientational order-disorder transition varying the angular noise η and identified the critical noise η_{c} for different v_{f}. Interestingly, both the species exhibit continuous transition for v_{f}<100v_{s} and discontinuous transition for v_{f}>100v_{s}. A new set of critical exponents is determined for the continuous transitions. However, the binary model is found to be nonuniversal as the values of the critical exponents depend on the velocity. The effect of interaction radius on the system behavior is also studied.

First-order transition in a percolation model with nucleation and preferential growth.

The spanning cluster properties of a percolation model with nucleation and preferential growth exhibit first-order transitions depending on the values of the growth parameter g_{0} and the initial seed concentration ρ. Except for the preferential growth of smaller clusters with a size-dependent growth probability of amplitude g_{0}, the model preserves all other criteria of the original percolation model. As ρ decreases starting from the percolation threshold p_{c} of the original percolation, a line of continuous transition encounters a coexistence region of percolative and nonpercolative large clusters. At sufficiently small values of ρ (≤0.05), the value of g_{0} exceeds p_{c} and generates compact spanning clusters leading to first-order discontinuous transitions.

Dissipative stochastic sandpile model on small-world networks: Properties of nondissipative and dissipative avalanches.

A dissipative stochastic sandpile model is constructed and studied on small-world networks in one and two dimensions with different shortcut densities ϕ, where ϕ=0 represents regular lattice and ϕ=1 represents random network. The effect of dimension, network topology, and specific dissipation mode (bulk or boundary) on the the steady-state critical properties of nondissipative and dissipative avalanches along with all avalanches are analyzed. Though the distributions of all avalanches and nondissipative avalanches display stochastic scaling at ϕ=0 and mean-field scaling at ϕ=1, the dissipative avalanches display nontrivial critical properties at ϕ=0 and 1 in both one and two dimensions. In the small-world regime (2^{-12}≤ϕ≤0.1), the size distributions of different types of avalanches are found to exhibit more than one power-law scaling with different scaling exponents around a crossover toppling size s_{c}. Stochastic scaling is found to occur for ss_{c}. As different scaling forms are found to coexist in a single probability distribution, a coexistence scaling theory on small world network is developed and numerically verified.

Crossover from rotational to stochastic sandpile universality in the random rotational sandpile model.

In the rotational sandpile model, either the clockwise or the anticlockwise toppling rule is assigned to all the lattice sites. It has all the features of a stochastic sandpile model but belongs to a different universality class than the Manna class. A crossover from rotational to Manna universality class is studied by constructing a random rotational sandpile model and assigning randomly clockwise and anticlockwise rotational toppling rules to the lattice sites. The steady state and the respective critical behavior of the present model are found to have a strong and continuous dependence on the fraction of the lattice sites having the anticlockwise (or clockwise) rotational toppling rule. As the anticlockwise and clockwise toppling rules exist in equal proportions, it is found that the model reproduces critical behavior of the Manna model. It is then further evidence of the existence of the Manna class, in contradiction with some recent observations of the nonexistence of the Manna class.

Critical properties of a dissipative sandpile model on small-world networks.

A dissipative sandpile model is constructed and studied on small-world networks (SWNs). SWNs are generated by adding extra links between two arbitrary sites of a two-dimensional square lattice with different shortcut densities ϕ. Three regimes are identified: regular lattice (RL) for ϕ≲2(-12), SWN for 2(-12)<ϕ<0.1, and random network (RN) for ϕ≥0.1. In the RL regime, the sandpile dynamics is characterized by the usual Bak, Tang, and Weisenfeld (BTW)-type correlated scaling, whereas in the RN regime it is characterized by mean-field scaling. On SWNs, both scaling behaviors are found to coexist. Small compact avalanches below a certain characteristic size s(c) are found to belong to the BTW universality class, whereas large, sparse avalanches above s(c) are found to belong to the mean-field universality class. A scaling theory for the coexistence of two scaling forms on a SWN is developed and numerically verified. Though finite-size scaling is not valid for the dissipative sandpile model on RLs or on SWNs, it is found to be valid on RNs for the same model. Finite-size scaling on RNs appears to be an outcome of super diffusive sand transport and uncorrelated toppling waves.

Flooding transition in the topography of toppling surfaces of stochastic and rotational sandpile models.

A continuous phase transition occurs in the topography of toppling surfaces of stochastic and rotational sandpile models when they are flooded with liquid, say water. The toppling surfaces are extracted from the sandpile avalanches that appear due to sudden burst of toppling activity in the steady state of these sandpile models. Though a wide distribution of critical flooding heights exists, a critical point is defined by merging the flooding thresholds of all the toppling surfaces. The criticality of the transition is characterized by power-law distribution of island area in the critical regime. A finite size scaling theory is developed and verified by calculating several new critical exponents. The flooding transition is found to be an interesting phase transition and does not belong to the percolation universality class. The universality class of this transition is found to depend on the degree of self-affinity of the toppling surfaces characterized by the Hurst exponent H and the fractal dimension D(f) of critical spanning islands. The toppling surfaces of different stochastic sandpile models are found to have a single Hurst exponent, whereas those of different rotational sandpile models have another Hurst exponent. As a consequence, the universality class of different sandpile models remains preserved within the same symmetry of the models.

Modeling the biomass growth and enzyme secretion by the white rot fungus Phanerochaete chrysosporium: a stochastic-based approach.

The white rot fungus Phanerochaete chrysosporium has been identified to be an environmentally useful microorganism for the degradation of various hazardous pollutants, mainly because of its ligninolytic enzyme system, particularly the lignin peroxidase (LiP) secreted by the fungus. In the present work, the behavior of the fungus in liquid medium due to variation in physico-chemical parameters, i.e., glucose concentration, nitrogen concentration, agitation, etc., was studied. Increment of the initial concentration of glucose in the medium increases the biomass growth and LiP activity, when cultured under controlled conditions. The biomass growth and LiP activity by the fungus was modeled following stochastic approach. The behavior of growth and enzyme activity of the fungus observed from the model were found to be in agreement with the experiments qualitatively.

Invasion of a sticky random solid: self-established potential gradient, phase separation, and criticality.

Invasion of a sticky random solid by an aqueous solution is modeled through a chemical reaction. In this reaction, the solid elements dissolve in the solution and redeposit back on the rough interface. A self-established potential gradient (SEPG) in the binding energy of the solid is developed spontaneously and the system gets phase separated into "hard" and "soft" solids. The solution profile is found drifted slowly into the solid by the SEPG with a constant velocity. The system tunes itself to the percolation threshold in the steady state. In the steady state, the system is found consisting of finite clusters of solution molecules followed by a path of redeposited solid as an invasion percolation cluster. A diffusive growth of the interface and the solution inside the solid is found to occur. The nonequilibrium steady state of this dynamical system is found critical and characterized by a power-law distribution of cluster size with an exponent approximately -2 .

Characteristics of deterministic and stochastic sandpile models in a rotational sandpile model.

Rotational constraint representing a local external bias generally has a nontrivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena. In order to study the effect of rotational bias in an out-of-equilibrium situation like self-organized criticality, a two state "quasideterministic" rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains. An extended set of critical exponents are estimated to characterize the avalanche properties at the nonequilibrium steady state of the model. The probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves. The model exhibits characteristics of both deterministic and stochastic sandpile models.

Directed self-avoiding walks in random media.

Two types of directed self-avoiding walks (SAW's), namely, three-choice directed SAW and outwardly directed SAW, have been studied on infinite percolation clusters on the square lattice in two dimensions. The walks on the percolation clusters are generated via a Monte Carlo technique. The longitudinal extension R(N) and the transverse fluctuation W(N) have been measured as a function of the number of steps N. Slight swelling is observed in the longitudinal direction on the random lattices. A crossover from shrinking to swelling of the transverse fluctuations is found at a certain length N(c) of the walks. The exponents related to the transverse fluctuations are seen to be unchanged in the random media even as the percolation threshold is reached. The scaling function form of the extensions are verified.